Theorem elcncf1di 15166

Description: Membership in the set of continuous complex functions from A to B. (Contributed by Paul Chapman, 26-Nov-2007.)

Hypotheses

Ref	Expression
elcncf1d.1	|- ( ph -> F : A --> B )
elcncf1d.2	|- ( ph -> ( ( x e. A /\ y e. RR+ ) -> Z e. RR+ ) )
elcncf1d.3	|- ( ph -> ( ( ( x e. A /\ w e. A ) /\ y e. RR+ ) -> ( ( abs ` ( x - w ) ) < Z -> ( abs ` ( ( F ` x ) - ( F ` w ) ) ) < y ) ) )

Assertion

Ref	Expression
elcncf1di	|- ( ph -> ( ( A C_ CC /\ B C_ CC ) -> F e. ( A -cn-> B ) ) )

Distinct variable groups:   x, w, y, A   w, B, x, y   w, F, x, y   ph, w, x, y   w, Z

Allowed substitution hints:   Z( x, y)

Proof of Theorem elcncf1di

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elcncf1d.1	. . 3 |- ( ph -> F : A --> B )
2		elcncf1d.2	. . . . . 6 |- ( ph -> ( ( x e. A /\ y e. RR+ ) -> Z e. RR+ ) )
3	2	imp 124	. . . . 5 |- ( ( ph /\ ( x e. A /\ y e. RR+ ) ) -> Z e. RR+ )
4		an32 562	. . . . . . . . 9 |- ( ( ( x e. A /\ w e. A ) /\ y e. RR+ ) <-> ( ( x e. A /\ y e. RR+ ) /\ w e. A ) )
5	4	anbi2i 457	. . . . . . . 8 |- ( ( ph /\ ( ( x e. A /\ w e. A ) /\ y e. RR+ ) ) <-> ( ph /\ ( ( x e. A /\ y e. RR+ ) /\ w e. A ) ) )
6		anass 401	. . . . . . . 8 |- ( ( ( ph /\ ( x e. A /\ y e. RR+ ) ) /\ w e. A ) <-> ( ph /\ ( ( x e. A /\ y e. RR+ ) /\ w e. A ) ) )
7	5, 6	bitr4i 187	. . . . . . 7 |- ( ( ph /\ ( ( x e. A /\ w e. A ) /\ y e. RR+ ) ) <-> ( ( ph /\ ( x e. A /\ y e. RR+ ) ) /\ w e. A ) )
8		elcncf1d.3	. . . . . . . 8 |- ( ph -> ( ( ( x e. A /\ w e. A ) /\ y e. RR+ ) -> ( ( abs ` ( x - w ) ) < Z -> ( abs ` ( ( F ` x ) - ( F ` w ) ) ) < y ) ) )
9	8	imp 124	. . . . . . 7 |- ( ( ph /\ ( ( x e. A /\ w e. A ) /\ y e. RR+ ) ) -> ( ( abs ` ( x - w ) ) < Z -> ( abs ` ( ( F ` x ) - ( F ` w ) ) ) < y ) )
10	7, 9	sylbir 135	. . . . . 6 |- ( ( ( ph /\ ( x e. A /\ y e. RR+ ) ) /\ w e. A ) -> ( ( abs ` ( x - w ) ) < Z -> ( abs ` ( ( F ` x ) - ( F ` w ) ) ) < y ) )
11	10	ralrimiva 2581	. . . . 5 |- ( ( ph /\ ( x e. A /\ y e. RR+ ) ) -> A. w e. A ( ( abs ` ( x - w ) ) < Z -> ( abs ` ( ( F ` x ) - ( F ` w ) ) ) < y ) )
12		breq2 4063	. . . . . 6 |- ( z = Z -> ( ( abs ` ( x - w ) ) < z <-> ( abs ` ( x - w ) ) < Z ) )
13	12	rspceaimv 2892	. . . . 5 |- ( ( Z e. RR+ /\ A. w e. A ( ( abs ` ( x - w ) ) < Z -> ( abs ` ( ( F ` x ) - ( F ` w ) ) ) < y ) ) -> E. z e. RR+ A. w e. A ( ( abs ` ( x - w ) ) < z -> ( abs ` ( ( F ` x ) - ( F ` w ) ) ) < y ) )
14	3, 11, 13	syl2anc 411	. . . 4 |- ( ( ph /\ ( x e. A /\ y e. RR+ ) ) -> E. z e. RR+ A. w e. A ( ( abs ` ( x - w ) ) < z -> ( abs ` ( ( F ` x ) - ( F ` w ) ) ) < y ) )
15	14	ralrimivva 2590	. . 3 |- ( ph -> A. x e. A A. y e. RR+ E. z e. RR+ A. w e. A ( ( abs ` ( x - w ) ) < z -> ( abs ` ( ( F ` x ) - ( F ` w ) ) ) < y ) )
16	1, 15	jca 306	. 2 |- ( ph -> ( F : A --> B /\ A. x e. A A. y e. RR+ E. z e. RR+ A. w e. A ( ( abs ` ( x - w ) ) < z -> ( abs ` ( ( F ` x ) - ( F ` w ) ) ) < y ) ) )
17		elcncf 15160	. 2 |- ( ( A C_ CC /\ B C_ CC ) -> ( F e. ( A -cn-> B ) <-> ( F : A --> B /\ A. x e. A A. y e. RR+ E. z e. RR+ A. w e. A ( ( abs ` ( x - w ) ) < z -> ( abs ` ( ( F ` x ) - ( F ` w ) ) ) < y ) ) ) )
18	16, 17	syl5ibrcom 157	1 |- ( ph -> ( ( A C_ CC /\ B C_ CC ) -> F e. ( A -cn-> B ) ) )

Syntax hints:   -> wi 4   /\ wa 104   e. wcel 2178   A.wral 2486   E.wrex 2487   C_ wss 3174   class class class wbr 4059   -->wf 5286   ` cfv 5290  (class class class)co 5967   CCcc 7958   < clt 8142   - cmin 8278   RR+crp 9810   abscabs 11423   -cn->ccncf 15157

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603  ax-cnex 8051

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-ral 2491  df-rex 2492  df-rab 2495  df-v 2778  df-sbc 3006  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-br 4060  df-opab 4122  df-id 4358  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-fv 5298  df-ov 5970  df-oprab 5971  df-mpo 5972  df-map 6760  df-cncf 15158

This theorem is referenced by:  elcncf1ii  15167  cncfmptc  15183  cncfmptid  15184  addccncf  15187  negcncf  15192